The first difficulty in this project is defining analogues of Lipschitz graphs in \(\mathbb {H}\) . Unlike the Euclidean case, we cannot define Lipschitz graphs as the images of Lipschitz maps from \(\mathbb {R}^2\)  to \(\mathbb {H}\)  or \(\mathbb {R}^3\)  to \(\mathbb {H}\) ; by a result of Ambrosio and Kirchheim [1]}, \(\mathcal {H}^3(f(\mathbb {R}^3))=0\)  for all Lipschitz functions \(f\colon \mathbb {R}^3 \rightarrow \mathbb {H}\) . Franchi, Serapioni and Serra Cassano [2]} introduced an intrinsic notion of Lipschitz graphs in Carnot groups which has been very influential in the development of sub-Riemannian geometric measure theory, see e.g. [3]}, [4]} and the references therein. Intrinsic Lipschitz graphs satisfy a cone condition which will be defined in Section REF . Moreover, they are 3–Ahlfors regular and thus the question of the \(L_2\) –boundedness of the Heisenberg Riesz transform on intrinsic Lipschitz graphs makes sense.
